📝 Abstract

We investigated the scaling and topology of engineered urban drainage networks (UDNs) in two cities, and further examined UDN evolution over decades. UDN scaling was analyzed using two power-law characteristics widely employed for river networks: (1) Hack’s law of length ( $L$)-area ( $A$) scaling [ $L \propto A^{h} $], and (2) exceedance probability distribution of upstream contributing area $(\delta)$ [ $P(A\geq \delta) \sim a \delta^{-\epsilon} $]. For the smallest UDNs ( $<2 \>\text{km}^2 $), length-area scales linearly ( $h\sim 1 $), but power-law scaling emerges as the UDNs grow. While $P(A\geq \delta)$ plots for river networks are abruptly truncated, those for UDNs display exponential tempering [ $P(A\geq \delta) \>\text{=}\> a \delta^{-\epsilon}\exp(-c\delta) $]. The tempering parameter $c$ decreases as the UDNs grow, implying that the distribution evolves in time to resemble those for river networks. However, the power-law exponent $\epsilon$ for large UDNs tends to be slightly larger than the range reported for river networks. Differences in generative processes and engineering design constraints contribute to observed differences in the evolution of UDNs and river networks, including subnet heterogeneity and non-random branching.

💡 Analysis

We investigated the scaling and topology of engineered urban drainage networks (UDNs) in two cities, and further examined UDN evolution over decades. UDN scaling was analyzed using two power-law characteristics widely employed for river networks: (1) Hack’s law of length ( $L$)-area ( $A$) scaling [ $L \propto A^{h} $], and (2) exceedance probability distribution of upstream contributing area $(\delta)$ [ $P(A\geq \delta) \sim a \delta^{-\epsilon} $]. For the smallest UDNs ( $<2 \>\text{km}^2 $), length-area scales linearly ( $h\sim 1 $), but power-law scaling emerges as the UDNs grow. While $P(A\geq \delta)$ plots for river networks are abruptly truncated, those for UDNs display exponential tempering [ $P(A\geq \delta) \>\text{=}\> a \delta^{-\epsilon}\exp(-c\delta) $]. The tempering parameter $c$ decreases as the UDNs grow, implying that the distribution evolves in time to resemble those for river networks. However, the power-law exponent $\epsilon$ for large UDNs tends to be slightly larger than the range reported for river networks. Differences in generative processes and engineering design constraints contribute to observed differences in the evolution of UDNs and river networks, including subnet heterogeneity and non-random branching.

📄 Content

1 Comparing Topology of Engineered and Natural Drainage Networks Soohyun Yang1,2, Kyungrock Paik2*, Gavan McGrath1,3, Christian Urich4,
Elisabeth Kruger1,5, Praveen Kumar6, and P. Suresh C. Rao1,7

1Lyles School of Civil Engineering, Purdue University, West Lafayette, IN, USA 2School of Civil, Environmental, and Architectural Engineering,
Korea University, Seoul, South Korea 3Ishka Solutions, Perth, Western Australia, Australia 4Civil Engineering Department, Monash University, Melbourne, Victoria, Australia 5Helmholtz Centre for Environmental Research - UFZ, Leipzig, Germany 6Department of Civil and Environmental Engineering,
University of Illinois at Urbana-Champaign, Urbana, IL, USA 7Agronomy Department, Purdue University, West Lafayette, IN, USA

*Corresponding author: Kyungrock Paik (paik@korea.ac.kr)

2 Abstract We investigated the scaling and topology of engineered urban drainage networks (UDNs) in two cities, and further examined UDN evolution over decades. UDN scaling was analyzed using two power-law characteristics widely employed for river networks: (1) Hack’s law of length (L)- area (A) scaling [L ∝ Ah], and (2) exceedance probability distribution of upstream contributing area (δ) [ ( ) ~ P A a ε δ δ − ≥ ]. For the smallest UDNs (< 2 km2), length-area scales linearly (h~1), but power-law scaling emerges as the UDNs grow. While ( ) P A δ ≥ plots for river networks are abruptly truncated, those for UDNs display exponential tempering [ ( ) exp( ) P A a c ε δ δ δ − ≥

− ]. The tempering parameter c decreases as the UDNs grow, implying that the distribution evolves in time to resemble those for river networks. However, the power-law exponent ε for large UDNs tends to be slightly larger than the range reported for river networks. Differences in generative processes and engineering design constraints contribute to observed differences in the evolution of UDNs and river networks, including subnet heterogeneity and non-random branching.

3 1 Introduction

Efficient drainage of urban landscapes is among the critical services provided to the citizens for avoiding flooding of streets and neighborhoods, and for maintaining flows to wastewater treatment plants (WWTPs). Here, of particular interest are urban drainage networks (UDNs) which include storm-water, sanitary sewers, and combined sanitary storm-water systems. UDNs are located below-ground at shallow depths and in close proximity to road networks [Blumensaat et al., 2012; Klinkhamer et al., 2017; Mair et al., 2012; Mair et al., 2017]. Our understanding about the network characteristics of engineered UDNs is limited with surprisingly little literature (e.g., Oh [2010]). An intriguing question in this regard is how the topology of UDNs compares with their natural analogs, i.e., river networks. UDNs, like rivers, involve gravity-driven and directed flows from the entire drainage area converging towards a WWTP. Many large cities have multiple outlets (e.g., combined sewer overflow outlets; several WWTPs) forming multiple sewer-sheds, whose boundaries may overlap several natural watersheds. UDNs consist of junctions and conduits, which correspond to nodes and edges, similar to confluences and reaches in river networks. Like stream orders in river networks, UDNs exhibit hierarchy of pipe- diameters for a range of designed maximum flows, and flows are directed towards an outlet. These structural and functional similarities prompted the application of river network hierarchical organization concepts to describe UDNs. Cantone and Schmidt [2011a, 2011b], Sitzenfrei et al. [2013], and Urich et al. [2010] classified the hierarchy of real/virtual sewer systems through Horton-Strahler ordering scheme [Strahler, 1957]. While river networks evolve through natural processes, UDNs are engineered networks designed to meet efficiently urban drainage requirements at the minimum cost. River networks drain large landscapes, up to the continental scales (~106 km2; e.g., Amazon; Congo; Nile; Rio de la Plata; Mississippi). In contrast, even the largest known UDNs drain much smaller urban sewer-sheds (≤ 103 km2) [USEPA, 2001].

Analogies and differences (see Table 1) lead us to explore differences in network topology of UDNs to those reported for river networks. It is well-established that river networks are fractal with self-similarity revealed through quantitative scaling relationships [Hack, 1957; Horton, 1945; Marani et al., 1994; Rigon et al., 1996; Rodríguez-Iturbe et al., 1992a; Tarboton et al., 1991; Tokunaga, 1978]. We examine here whether UDNs and river networks share scaling properties reported for river networks, given similarities in their functions in landscape drainage, and despite differences between engineered versus natural networks.

We investigate here the functional organization and scaling of UDNs in terms of their topological features, and examine how scaling patterns change as

This content is AI-processed based on ArXiv data.